Les variations du champ magnétique terrestre sont documentées par les observatoires au sol et les satellites en orbite basse, pour des échelles de temps de l'année au siècle.Sur ces périodes, la dynamique du noyau externe -- là où est principalement généré le champ magnétique -- est fortement influencée par la rotation terrestre, qui tend à imposer une invariance dans la direction parallèle à l'axe de rotation.Dans cette thèse, j'étudie un modèle s'appuyant sur cette hypothèse de bidimensionnalité du champ de vitesse, le modèle quasi-géostrophique.Je présente une nouvelle dérivation de ce modèle par une approche variationnelle, plus adaptée aux fortes pentes des frontières du domaine sphérique.Je présente une étude modale des ondes hydro-magnétiques, qui pour la première fois prend en compte l'impact d'un champ magnétique imposé non-zonal.Deux groupes d'ondes magnéto-hydrodynamiques apparaissent alors : les ondes magnéto-Coriolis (centennales) et les ondes d'Alfvéen de torsion (interannuelle).Je décris l'évolution des ondes à mesure que l'on intensifie l'effet de la rotation, jusqu'à atteindre des paramètres géophysiques.Je discute également dans quel mesure une version du modèle quasi-géostrophique où la force de Lorentz est représentée par des produits quadratiques du champ magnétique est adapté pour l'interprétation de calculs numériques tridimensionnels de la dynamo. J'observe que pour les paramètres aujourd'hui accessibles à ces calculs, les forces magnétiques sont faibles. À long terme, nous espérons utiliser le modèle quasi-géostrophique dans le contexte de l'assimilation de données satellitaires. / Variations of the Earth's magnetic field are documented by ground observatories and low-orbiting satellites, for time scales from year to century.On such periods, dynamics of the outer core -- where the creation of the magnetic field takes place -- is strongly influenced by the Earth rotation, which tends to impose invariance of the flow in the direction parallel to the rotation axis.In this thesis report, I study a model based on this bi-dimensional velocity field hypothesis, the quasi-geostrophic model.I present a new mathematical formulation of this model through a variational approach, better suited to steep slopes on the boundaries of the spherical domain.I present a modal study of hydromagnetic waves, taking into account for the first time the impact of a non-zonal imposed magnetic field.Two groups of hydromagnetic waves are present : centennial magneto-Coriolis waves and interannual torsional Alfvén waves.I describe evolution of those waves as the effect of rotation is intensified until Earth-like parameters are reached.I also discuss in what measure an other version of the quasi-geostrophic model, where Lorentz force is represented by quadratic products of the magnetic field, can be adapted to understand tridimensional dynamo numerical simulations.I observe that for parameters available today, magnetic forces are weak.In the future, we hope to use the Quasi-geostrophic model in the context of satellite data assimilation.

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In this thesis, quantum-mechanical calculations within density-functional theory on metallic systems are presented. The overarching goal has been to investigate effects of disorder. In particular, one of the properties investigated is the bindingenergy shifts for core electrons in binary alloys using different theoretical methods. These methods are compared with each other and with experimental results. One such method, the so-called Slater-Janak transition state method relies on the assumption that the single-particle eigenvalues within density-functional theory are linear functions of their respective occupation number. This assumption is investigated and it is found that while the eigenvalues to a first approximation show linear behavior, there are also nonlinearities which can influence the core-level binding energy shifts. Another area of investigation has been iron based alloys at pressures corresponding to those in the Earth’s inner core. This has been done for the hexagonal close packed and face entered cubic structures. The effects of alloying iron with magnesium and nickel on the equation of state as well on the elastic properties have been investigated. The calculations have shown that the hexagonal close packed structure in FeNi is more isotropic than the face-centered cubic structure, and that adding Mg to Fe has a large impact on the elastic properties. Finally, the effects of disorder due to thermal motion of the atoms have been investigated through ab-initio molecular dynamics simulations. Within the limits of this method and the setup, it is found that the face-centered cubic structure of molybdenum can be dynamically stabilized at high temperature, leading to a metastable structure, on the average. The dynamical stabilization of face-centered cubic molybdenum also rendered it possible to accurately calculate the lattice stability relative to the body-centered cubic phase. Inclusion of temperature effects for the lattice stability using ab-initio molecular dynamics simulations resolves the disagreement between ab-initio calculations and thermochemical methods.